Hochschild cohomology and Atiyah classes

Damien Calaque (1) , Michel van Den Bergh

1 ICJ - Institut Camille Jordan

In this paper we provide a proof of a result announced by Kontsevich that the HKR-morphism, twisted by the square root of the Todd genus, gives an isomorphism between the Hochschild cohomology of a smooth algebraic variety and the cohomology of poly-vector fields, both considered as Gerstenhaber algebras. Our proof is set in the framework of Lie algebroids and so applies in more general settings as well.

Domaines: K-théorie et homologie [math.KT]; Algèbres quantiques [math.QA]

Damien Calaque, Michel van Den Bergh. Hochschild cohomology and Atiyah classes. Advances in Mathematics, 2010, 224 (5), pp.1839-1889. ⟨10.1016/j.aim.2010.01.012⟩. ⟨hal-00167742⟩